After failing to score in the first 49 minutes against Mike Smith (30 saves), Washington tallied three times in a span of 5 minutes, 2 seconds. Karl Alzner started the surge with a goal at the 9:45 mark. Just 32 seconds later, Brooks Laich tied it on a feed from Jason Chimera.
When Washington got just its second power-play opportunity of the night, David Moss tripping Dmitry Orlov, the Capitals cashed in. Smith failed to control a slap shot by Nicklas Backstrom and it rebounded to Brouwer who scored his seventh goal in his last eight games.
The win snapped Washington's three-game losing streak. The Capitals remained in 10th place in the Eastern Conference standings, one point behind Columbus and Detroit, who are tied for the eighth and final playoff spot.
''We know that we're at a point where our season is on the line every single game,'' Brouwer said. ''That should be enough motivation in here and it was tonight.''
It was a successful Washington debut for goalie Jaroslav Halak, who made 31 saves. Halak was acquired from Buffalo in a deadline deal on Wednesday.
''It was really weird, wearing that jersey, looking down, seeing the red color,'' Halak said. ''The first half of the game, I was really nervous. I wanted to obviously leave a good impression on fans on my teammates and everybody in the organization.''
The victory ended an eventful night for the Capitals. Less than two hours before the game, Washington signed 2010 first-round draft choice Kuznetsov, ending a long pursuit of the 21-year-old forward, who has played the last four years in Russia's Kontinental Hockey League. He could make his NHL debut Monday night when the Capitals host the Pittsburgh Penguins.
''He doesn't know this hockey and we don't want to put him in a situation that he can't handle right now,'' Washington coach Adam Oates said. ''But in saying that, if we can spot him in the lineup it could be a nice spark for us. He's a very talented man.''
''It seemed like we stopped playing the way we played in the first and the second,'' Vrbata said. ''When you have a 2-0 lead going into the third, you have to find a way to win, and we didn't.''
The Coyotes took the lead with 62 seconds left in the first period when Washington lost the puck in the neutral zone as it began a transition play. McMillan turned the mistake into a breakaway goal, his first in the NHL since scoring in the 2011 playoffs for Anaheim.
It might have been the second goal for McMillan. Midway through the period, he appeared to score off a rebound. But officials ruled that McMillan simultaneously pushed the puck and Halak's pad across the goal line.
In the second period, Phoenix made it 2-0 as Vrbata redirected a shot from the point by Michael Stone.
After losing 3-0 on Thursday at Boston, Washington appeared destined for a second straight shutout until Alzner scored from deep as Yandle appeared to screen the view of Smith - starting the big comeback. It was just the second goal this season for the defenseman.
''Maybe it had something to do with the fact that it was me who scored it and not a guy who normally scores, but people seemed really up after that,'' Alzner said. ''It was the perfect momentum swing and the perfect line to come out right after that. It was a perfect storm for us.''
Notes: Phoenix forward Martin Erat was booed many times when he touched the puck. Erat was dealt by Washington this week after scoring one goal for the team. In November, Erat asked to be traded, complaining about a lack of playing time.